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What is the probability that a large random matrix has no real eigenvalues? (English) Zbl 1375.60019
Summary: We study the large-\(n\) limit of the probability \(p_{2n,2k}\) that a random \(2n \times 2n\) matrix sampled from the real Ginibre ensemble has \(2k\) real eigenvalues. We prove that \[ \lim_{n\longrightarrow \infty} \frac{1}{\sqrt 2n} \log p_{2n,2k} = \lim_{n\longrightarrow \infty} \frac{1}{\sqrt 2n} \log p_{2n,0} = - \frac{1}{\sqrt 2\pi} \zeta \left (\frac 32\right ) , \] where \(\zeta\) is the Riemann zeta-function. Moreover, for any sequence of nonnegative integers \((k_n)_{n\geq 1}\), \[ \lim_{n\longrightarrow \infty} \frac{1}{\sqrt 2n}\log p_{2n,2k_n} = - \frac{1}{\sqrt 2\pi} \zeta \left (\frac 32\right), \] provided \(\displaystyle \lim_{n\to\infty}\left (n^{-1/2}\log(n)\right)k_n=0\).

MSC:
60B20 Random matrices (probabilistic aspects)
60F10 Large deviations
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